.gif)
Defect (geometry)
Encyclopedia
In geometry
, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.
Classically the defect arises in two ways:
and the excess arises in one way:
In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).
In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem
.
is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. If the sum of the angles exceeds a full circle, as occurs in some vertices of most (not all) non-convex polyhedra, then the defect is negative. If a polyhedron is convex, then the defects of all of its vertices are positive.
The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angle
s of the cells
at a peak falls short of a full circle.
(According to the Oxford English Dictionary
, one of the senses of the word "defect" is "The quantity or amount by which anything falls short; in Math. a part by which a figure or quantity is wanting or deficient.")
s meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles is 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.
The same procedure can be followed for the other Platonic solid
s
to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4π radians). The polyhedron need not be convex.
A generalization says the number of circles in the total defect equals the Euler characteristic
of the polyhedron. This is a special case of the Gauss–Bonnet theorem
which relates the integral of the Gaussian curvature
to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the Gaussian curvature is zero and the Gaussian curvature at a vertex is equal to the defect there.
This can be used to calculate the number V of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.
It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider a cube
where one face is replaced by a square pyramid
: this elongated square pyramid
is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is non-convex, but the defects remain the same and so are all positive.
Negative defect indicates that the vertex resembles a saddle point
, whereas positive defect indicates that the vertex resembles a local maximum or minimum.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.
Classically the defect arises in two ways:
- the defect of a vertex of a polyhedron;
- the defect of a hyperbolic triangleHyperbolic triangleIn mathematics, the term hyperbolic triangle has more than one meaning.-Hyperbolic geometry:In hyperbolic geometry, a hyperbolic triangle is a figure in the hyperbolic plane, analogous to a triangle in Euclidean geometry, consisting of three sides and three angles...
;
and the excess arises in one way:
- the excess of a spherical triangle.
In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).
In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem
Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
.
Defect of a vertex
The defect of a vertex of a polyhedronPolyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. If the sum of the angles exceeds a full circle, as occurs in some vertices of most (not all) non-convex polyhedra, then the defect is negative. If a polyhedron is convex, then the defects of all of its vertices are positive.
The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angle
Dihedral angle
In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
s of the cells
Cell (geometry)
In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
at a peak falls short of a full circle.
(According to the Oxford English Dictionary
Oxford English Dictionary
The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...
, one of the senses of the word "defect" is "The quantity or amount by which anything falls short; in Math. a part by which a figure or quantity is wanting or deficient.")
Examples
The defect of any of the vertices of a regular dodecahedron (in which three regular pentagonPentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
s meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles is 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.
The same procedure can be followed for the other Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s
| Shape | Number of vertices | Polygons meeting at each vertex | Defect at each vertex | Total defect |
|---|---|---|---|---|
| tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... |
4 | Three equilateral triangles |  |
 |
| octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... |
6 | Four equilateral triangles |  |
 |
| cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... |
8 | Three squares |  |
 |
| icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... |
12 | Five equilateral triangles |  |
 |
| dodecahedron | 20 | Three regular pentagons |  |
 |
Descartes' theorem
Descartes' theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphicHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4π radians). The polyhedron need not be convex.
A generalization says the number of circles in the total defect equals the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of the polyhedron. This is a special case of the Gauss–Bonnet theorem
Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
which relates the integral of the Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the Gaussian curvature is zero and the Gaussian curvature at a vertex is equal to the defect there.
This can be used to calculate the number V of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.
A potential error
 |
 |
It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
where one face is replaced by a square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
: this elongated square pyramid
Elongated square pyramid
In geometry, the elongated square pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base...
is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is non-convex, but the defects remain the same and so are all positive.
Negative defect indicates that the vertex resembles a saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
, whereas positive defect indicates that the vertex resembles a local maximum or minimum.